16.6 Parametric Surfaces and Their Areas

Claudia Castro-Castro
Math 283 Spring 2020

Instructions:

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Outline

  • The following topics will be covered in this section:
    • Parametric representation for a surface
    • Tangent plane to a surface
    • Surface area

Parametric representation for a surface

  • A surface is described by a vector function \[ \mathbf{\vec{r}}(u, v) = x(u,v)\mathbf{\hat{i}}+y(u,v)\mathbf{\hat{j}}+z(u,v)\mathbf{\hat{k}} \] of two parameters \( u \) and \( v \)
  • \[ \begin{align} x&=x(u,v)\\ y&=y(u,v)\\ z&=z(u,v) \end{align} \]

  • DEFINITION Given a parameterization of the surface \[ \mathbf{\vec{r}}(u, v) = x(u,v)\mathbf{\hat{i}}+y(u,v)\mathbf{\hat{j}}+z(u,v)\mathbf{\hat{k}} \] the parameter domain is the set of points in the \( uv \)-plane that can be substituted into \( \mathbf{\vec{r}} \)

Example: Graphs of functions \( z=f(x,y) \)

  • Let \[ x(u,v) =u, \qquad y(u,v) =v,\qquad z(u,v)=f(u,v) \]
  • Example: \( z=x^2+y^2 \)
  • \[ x(u,v) =u, \qquad y(u,v) =v,\qquad z(u,v)=u^2+v^2 \]
  • \[ \begin{align} \mathbf{\vec{r}}(u, v) &= u\mathbf{\hat{i}}+v\mathbf{\hat{j}}+(u^2+v^2)\mathbf{\hat{k}}\\ &or\\ &= \langle u,v,u^2+v^2\rangle \end{align} \]
  • Parameter domain \[ D=[-1,1]\times[-1,1] \]
Paraboloid defined on a rectangle

  • Parameter domain \[ D=\{(u,u)| \;u^2+v^2 \leq 1\} \]
Paraboloid defined on a disk

Example: Cones

  • \[ \mathbf{\vec{r}_1}(u, v)= \langle v, u, \sqrt{u^2+v^2}\rangle \]

  • Parameter domain \[ D=\{(u,v)|\; u^2+v^2\leq 1 \} \]
Circular cylinder



  • \[ \mathbf{\vec{r}_2}(u, v)= \langle v\cos u, v\sin u, v\rangle \]

  • Parameter domain \[ \begin{align} D&= \{(u,v)|\; 0 \leq u \leq 2\pi,\; 0\leq v\leq 1 \}\\ &=[0,2\pi]\times[0,1] \end{align} \]

  • \( \star \) Parameterizations are not unique \( \star \)
Circular cylinder

Example: Plane

Plane that passes through a point contains two non-parallel vectors

  • \( P(1,1,1), v_1=(-2,1,0), v_2=(2,-3,1) \)
Plane that passes through a point contains two non-parallel vectors

Example: Sphere

Sphere center at origin

  • Representation in spherical coordinates \[ \begin{align} \rho&=\text{constant}\\ u&=\theta\\ v&=\phi \end{align} \]
  • Then \[ \mathbf{\vec{r}}(u, v)= \rho\cos u \sin v\mathbf{\hat{i}}+\rho\sin u \sin v\mathbf{\hat{j}}+\rho \cos v\mathbf{\hat{k}} \]

  • Parameter domain \[ \begin{align} D&=\{(u, v)| \;0\leq u \leq 2\pi,\; 0\leq v \leq \pi\}\\ &=[0,2\pi]\times[0,\pi] \end{align} \]

Example: Cylinders

Circular cylinder

\[ \mathbf{\vec{r}}(u, v)= \langle 2 \cos u,2 \sin u, v\rangle\\ D=[0,2\pi]\times[-3,3] \]

Elliptic cylinder


\[ \mathbf{\vec{r}}(u, v)= \langle 2 \cos u,5 \sin u, v\rangle\\ D=[0,2\pi]\times[-4,4] \]
Parabolic cylinder

\[ \mathbf{\vec{r}}(u, v)= \langle u, v, v^2\rangle\\ D=[-2,2]\times[-2,2] \]

Example: Other surfaces

Spiral Spiral

Molusc shell Molusc shell

Torus/Donut Donut

Möbius strip Möbius

Points on surface

Surface and its parameter domain

Surface area

  • Suppose that a surface \( S \) has a vector equation \[ \mathbf{\vec{r}}(u, v) = x(u,v)\mathbf{\hat{i}}+y(u,v)\mathbf{\hat{j}}+z(u,v)\mathbf{\hat{k}}\\ (u,v)\in D \]
Surface and its parameter domain

  • Slice:
    Divide \( D \) into rectangles \( R_{ij} \)
    the patch \( S_{ij} \) corresponds to \( R_{ij} \)

Approximating the area of the patch with area of parallelogram

  • Approximate
    • The patch \( S_{ij} \) by the parallelogram determined by vectors
    • The edges of the patch that meet at \( P_{ij} \) by the vectors \[ \mathbf{\vec{r}}^*_u\; \Delta u_i\\ \mathbf{\vec{r}}^*_v\; \Delta v_j \]
    • Area of patch with cross product \[ A(S_{ij})\approx || \left(\mathbf{\vec{r}}^*_u\; \Delta u_i \right) \times \left(\mathbf{\vec{r}}^*_v\; \Delta v_j \right) || \]

Surface area cont'd

Approximating the area of the patch with area of parallelogram

  • Approximate
    • Area of patch with cross product \[ A(S_{ij})\approx ||\mathbf{\vec{r}}^*_u \times\mathbf{\vec{r}}^*_v || \Delta u_i \Delta v_j \]
  • Sum and take limit
  • \[ A(S)\approx \sum_{i}\sum_j ||\mathbf{\vec{r}}^*_u \times\mathbf{\vec{r}}^*_v || \Delta u_i \Delta v_j \]
  • \( \Delta u_i \rightarrow 0 \) and \( \Delta v_j \rightarrow 0 \)

    Surface area \[ A(S)=\iint\limits_D ||\mathbf{\vec{r}}_u \times \mathbf{\vec{r}}_v||dA \] \( D \) is the parameter domain

Example: surface area

  • Find the surface area of the part of the paraboloid \( z = f(x,y)= x^2 +y^2 \) that lies under the plane \( z=1 \)
Vector plot of rotating field

  • \[ A(S)=\iint\limits_D ||\mathbf{\vec{r}}_u \times \mathbf{\vec{r}}_v||dA \]
  • Parameterization \[ \mathbf{\vec{r}}(u, v)= \langle u,v,f(u,v)\rangle\quad D=\{(u,v)|\; u^2+v^2\leq 1 \} \]
  • Derive \[ \mathbf{\vec{r}}_u = \langle 1,0,\frac{\partial f}{\partial u} \rangle \]
  • \[ \mathbf{\vec{r}}_v = \langle 0,1,\frac{\partial f}{\partial v} \rangle \]
  • \[ \mathbf{\vec{r}}_u \times \mathbf{\vec{r}}_v=\left|\left(\begin{array}{ccc} \mathbf{\hat{{i}}} & \hat{{\mathbf{j}}} & \hat{{\mathbf{k}}}\\ 1 & 0 & \frac{\partial f}{\partial u}\\ 0 & 1 & \frac{\partial f}{\partial v}\\ \end{array}\right)\right| \]
  • \[ \mathbf{\vec{r}}_u \times \mathbf{\vec{r}}_v=- \frac{\partial f}{\partial u} \mathbf{\hat{{i}}} + \frac{\partial f}{\partial v} \mathbf{\hat{{j}}}+1\mathbf{\hat{k}} \]
  • \[ ||\mathbf{\vec{r}}_u \times \mathbf{\vec{r}}_v||=\sqrt{\left(\frac{\partial f}{\partial u} \right)^2+\left(\frac{\partial f}{\partial v} \right)^2 +1} \]

Example: surface area

Vector plot of rotating field

  • \[ ||\mathbf{\vec{r}}_u \times \mathbf{\vec{r}}_v||=\sqrt{\left(\frac{\partial f}{\partial u} \right)^2+\left(\frac{\partial f}{\partial v} \right)^2 +1} \]
  • Since \( f(u,v)=u^2+v^2 \) \[ \frac{\partial f}{\partial u} = 2u\qquad \frac{\partial f}{\partial v}=2v \]
  • Set up and evaluate \[ \begin{align} A(S) &= \iint\limits_D \sqrt{(2u)^2+(2v)^2+1}= \iint\limits_D \sqrt{4(u^2+v^2)+1}dA\end{align} \]
  • Using polar coordinates \[ \begin{align} A(S) & = \int_0^{2\pi} \int_0^1 \sqrt{4r^2+1}\;rdrd\theta\end{align} \]
  • \[ \begin{align} A(S)&= 2\pi \left( \frac{1}{8}\right)\left(\frac{2}{3}\right) \left(4r^2+1\right)^{3/2}\bigg\vert_0^1\\ &= \frac{\pi}{6}\left(5^{3/2} -1 \right)\end{align} \]

Tangent planes

Diagram of a surface and its parameter domain

Example: tangent planes

Diagram of a surface and its parameter domain

Final remarks